On a class of nonlinear Schrödinger equations. (English) Zbl 0763.35087

This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations \[ i\hslash\psi_ t=-{\hslash^ 2\over 2m}\Delta\psi+V(x)\psi-\gamma|\psi|^{p-1}\psi, \tag{1} \] where \(x\in \mathbb R^ n\), \(1<p<(n+2)/(n-2)\). Making a standing wave ansatz the problem reduces to that of studying the semilinear elliptic equation. In fact, a more general semilinear elliptic PDE \[ -\Delta V+b(x)v=f(x,v),\quad x\in \mathbb R \tag{2} \] is considered. Using “mountain pass” and comparison arguments, the author gets existence of nontrivial solutions \(u\in W^{1,2}(\mathbb R^ n)\) for (2), under some additional assumptions on \(b(x)\) and \(f(x,u)\).


35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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